I0.60] oI o

The A matrix has been transformed to the identity matrix I and the b vector has been transformed to the solution vector, x. Thus, xT = [0.60 1.00 0.40].

The inverse of a square matrix A is the matrix A -I such that AA -1 = A-I A = I. Gauss-Jordan elimination can be used to evaluate the inverse of matrix A by augmenting A with the identity matrix I and applying the Gauss-Jordan algorithm. The transformed A matrix is the identity matrix I, and the transformed identity matrix is the matrix inverse, A-I. Thus, applying Gauss-Jordan elimination yields

(1.1 09)

The Gauss-Jordan elimination procedure, in a format suitable for programming on a computer, can be developed to solve Eq. (1.109) by modifying the Gauss elimination procedure presented in Section 1.3.e. Step I is changed to augment the n x n A matrix with the n x n identity matrix, I. Steps 2 and 3 of the procedure are the same. Before performing Step 3, the pivot element is scaled to unity by dividing all elements in the row by the pivot element. Step 3 is expanded to perform elimination above the pivot element as well as below the pivot element. At the conclusion of step 3, the A matrix has been transformed to the identity matrix, I, and the original identity matrix, I, has been transformed to the matrix inverse, A-I.